In vehicles of recent years, a control technique of enabling the vehicle to travel with inertia with the engine stopped, or to stop the engine when temporarily stopping the vehicle is being developed if a request for a driving force is not made by a driver during traveling of the vehicle in an aim to improve fuel economy, to reduce the emission amount of the exhaust gas, and the like. In this case, a starter, which is an engine starting device, is driven to start the engine without the driver carrying out the starting operation of the engine when a predetermined condition is satisfied with the engine stopped.
However, when performing control to stop the engine during traveling of the vehicle, the usage frequency of the starter becomes high and the endurance of the starter tends to lower easily, and thus the control of the vehicle is differed according to the usage state of the starter in some conventional vehicles. For example, in an electronic control device described in Patent Literature 1, a driving time of the starter is accumulated, and the control to stop the engine is prohibited when the accumulated driving time reaches a predetermined time to suppress damages of the starter and the gears.